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PreprintCC BY 4.0arXiv

Only finitely many modular Jacobians are supersingular modulo a given prime

  • Aarya J. Kumar
  • Sargam Mondal
  • Erick Ross
  • Hui Xue
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Abstract

In this paper, we study supersingularity of modular Jacobians J0(N) and abelian varieties Af of GL2-type from both the vertical perspective (where the prime p is fixed, and the level N varies) and the horizontal perspective (where the newform f is fixed, and the prime p varies). Vertically, we prove that J0(N) is supersingular modulo a given prime p for only finitely many levels N. Horizontally, we show that for sufficiently large p, the reduction of Af modulo p is supersingular if and only if it isogenous to a power of a supersingular elliptic curve.

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Cite

BibTeXRIS
BibTeX
@misc{kumar2026only,
  title = {{Only finitely many modular Jacobians are supersingular modulo a given prime}},
  author = {Aarya J. Kumar and Sargam Mondal and Erick Ross and Hui Xue},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17819},
  eprint = {2609.17819},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17819},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Only finitely many modular Jacobians are supersingular modulo a given prime
AU  - Aarya J. Kumar
AU  - Sargam Mondal
AU  - Erick Ross
AU  - Hui Xue
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.17819
UR  - https://arxiv.org/abs/2609.17819
AB  - In this paper, we study supersingularity of modular Jacobians J_0(N) and abelian varieties A_f of GL_2-type from both the vertical perspective (where the prime p is fixed, and the level N varies) and the horizontal perspective (where the newform f is fixed, and the prime p varies). Vertically, we prove that J_0(N) is supersingular modulo a given prime p for only finitely many levels N. Horizontally, we show that for sufficiently large p, the reduction of A_f modulo p is supersingular if and only if it isogenous to a power of a supersingular elliptic curve.
ER  -